BUG: Inconsistent dtype with GroupBy for StrDtype and all missing values

[WillAyd]

Pandas version checks

• I have checked that this issue has not already been reported.

• I have confirmed this bug exists on the latest version of pandas.

• I have confirmed this bug exists on the main branch of pandas.

Reproducible Example

>>> df = pd.DataFrame({"a": ["a"] * 3, "b": pd.Series([None] * 3, dtype=pd.StringDtype(na_value=np.nan))})
>>> df
   a    b
0  a  NaN
1  a  NaN
2  a  NaN
>>> df.groupby("a").sum()
   b
a   
a  0
>>> df.groupby("a").sum().dtypes
b    str
dtype: object
>>> df.groupby("a").min()
    b
a    
a NaN
>>> df.groupby("a").min().dtypes
b    float64
dtype: object

Issue Description

The sum reduction return type is partially discussed in #60229 but I didn't see anything for min

Note that this discrepancy is the root cause of the test failure shown at

pandas/pandas/tests/resample/test_resampler_grouper.py

Line 465 in dec6eb2

@pytest.mark.xfail(using_string_dtype(), reason="TODO(infer_string)")

@rhshadrach

Expected Behavior

I think in all cases here we should still be returning a str type.

Installed Versions

'3.0.0.dev0+1824.g8d6d29cac3.dirty'

[rhshadrach]

Thanks @WillAyd - taking this up now. I also checked out all other reductions, it appears only min and max have incorrect behavior.

One I'll note is that idxmin and idxmax return float NaN, but I think this is expected. This is because these are returning index values which are integers. When one changes the index to string, we do indeed return string dtype as well.

pd.set_option("infer_string", True)
df = pd.DataFrame({"a": "a", "b": pd.array(3*[None], dtype="str")}, index=["x", "y", "z"])
print(df.groupby("a").idxmin())
#      b
# a     
# a  NaN

print(df.groupby("a").idxmin().dtypes)
# b    str
# dtype: object